A120854 Matrix log of A117939, read by rows, consisting only of 0's, 3's and signed 2's.

0, 2, 0, 3, -2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 3, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, -2, 0, 2, 0, 0, 0, 3, 0, 0, -2, 3, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 3, -2, 0, 0, 0, 0, 0

Offset: 0

Author

Paul D. Hanna, Jul 08 2006

Keywords

• easy
• sign
• tabl

Comments

The number of nonzero elements in row n equals A053735(n), the sum of ternary digits of n. Row sums are A120855(n) = 2*A062756 + A081603(n), where A062756(n) = number of 1's in ternary expansion of n and A081603(n) = number of 2's in ternary expansion of n. Triangle A117939 is related to partitions of n into powers of 3 and is the matrix square of A117947, where A117947(n,k) = balanced ternary digits of C(n,k) mod 3, also A117947(n,k) = L(C(n,k)/3) where L(j/p) is the Legendre symbol of j and p.

Examples

			Triangle begins:
0;
2, 0;
3,-2, 0;
2, 0, 0, 0;
0, 2, 0, 2, 0;
0, 0, 2, 3,-2, 0;
3, 0, 0,-2, 0, 0, 0;
0, 3, 0, 0,-2, 0, 2, 0;
0, 0, 3, 0, 0,-2, 3,-2, 0;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0; ...
Matrix exponentiation gives A117939:
1;
2, 1;
1,-2, 1;
2, 0, 0, 1;
4, 2, 0, 2, 1;
2,-4, 2, 1,-2, 1;
1, 0, 0,-2, 0, 0, 1;
2, 1, 0,-4,-2, 0, 2, 1;
1,-2, 1,-2, 4,-2, 1,-2, 1; ...
and A117939 is the matrix square of A117947:
1;
1, 1;
1,-1, 1;
1, 0, 0, 1;
1, 1, 0, 1, 1;
1,-1, 1, 1,-1, 1;
1, 0, 0,-1, 0, 0, 1;
1, 1, 0,-1,-1, 0, 1, 1;
1,-1, 1,-1, 1,-1, 1,-1, 1; ...
		

Crossrefs

Cf. A117939, A117947; A120855 (row sums), A062756, A081603, A053735.

Programs

PARI

/* Generated as the Matrix LOG of A117939: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,(binomial(r-1,c-1)+1)%3-1)^2, L=sum(i=1,#M,-(M^0-M)^i/i));return(L[n+1,k+1])

PARI

/* Generated as the Ternary Fractal: */ T(n,k)=local(r=n,c=k,s=floor(log(n+1)/log(3))+1,u=vector(s),v=vector(s),d,e); if(n<=k,0,if(n<3&k<3,[0,0,0;2,0,0;3,-2,0][n+1,k+1], for(i=1,#u,u[i]=r%3;r=r\3);for(i=1,#v,v[i]=c%3;c=c\3); d=0;for(i=1,#v,if(u[i]!=v[i],d+=1;e=i));if(d==1,T(u[e],v[e]),0)))

Formula

Ternary fractal, T(3*n,3*k) = T(n,k), defined by: T(n,k) = 0 if n<=k or when more than 1 digit differs between the ternary expansions of n and k; else T(n,k) = T(m,j) where the only ternary digits of n, k, that differ are m, j, respectively and T(1,0)=2, T(2,1)=-2, T(2,0)=3.